A novel method is described of designing lenses for general illumination tasks that are not circularly symmetric. The crux of the method is the specification of the illumination task by a grid on the unit sphere of directions. This grid, or tessellation, has cells which vary in solid angle such that each encompasses the same luminous flux: high intensity corresponds to small cells, and vice versa. Another grid, having the same topology and number of cells, is formed according to the intensity distribution of the source. An illumination lens must then transform the source distribution into the task distribution, via one or more refractions. Thus the direction vector of a cell in the source grid must be redirected into that of the corresponding cell in the task grid. Snell's law in vector form enables the derivation of a corresponding surface normal vector, or sequence of normal vectors, that will accomplish this redirection. Extrinsic differential geometry is then used to generate a lens surface having, as closely as possible, this distribution of surface normals, which must be irrotational to generate a smooth surface. This class of lenses has only recently become producible due to the advent of electric-discharge machining for the shaping of non-rotationally symmetric injection molds for plastic lenses.